Reviews

Information about the First Edition:

The book is hardcover, 250 pages, published in 1992 by Addison-Wesley,
Advanced Book Program,
Reading, MA, in the
Santa Fe Institute Studies in the Sciences of Complexity,
Reference Volume I. ISBN: 0-201-55740-1.
For information at SFI
click HERE.

Related topics

Discrete Dynamics Lab (DDLab), the latest software and
operating manual
for studying discrete dynamical networks, running on Unix, Linux and Irix as
well as DOS.
This supersedes the software described in the book.

Attractor Basins of Discrete Networks, Wuensche's DPhil thesis written
after the book was published.
The thesis developments the ideas presented in
the book, generalizes the methods to Boolean networks,
and describes applications in neural and genetic networks.

Back Cover Text

The Global Dynamics of Cellular Automata
introduces a powerful new perspective for the study of discrete
dynamical systems. After first looking at the unique trajectory of a
system's future, an algoritm is presented that directly computes the
multiple merging trajectories of the systems past. A given cellular
automaton will "crystallize" state space into a set of basins of
attraction that typically have a topology of trees rooted on attractor
cycles. Portraits of these objects are made accessible through computer
generated graphics. The "Atlas" presents a complete class of such
objects, and is inteded , with the accompanying software, as an aid to
navigation into the vast reaches of rule behaviour space. The book will
appeal to students and researchers interested in cellular automata,
complex systems, computational theory, artificial life, neural networks,
and aspects of genetics.

Foreword - by Chris Langton

There are a wide variety of methods for representing the behavior of
dynamical systems. Perhaps the most familiar representation method is
the traditional time-series plot, in which some observable variable of
the system (e.g., angular position) is plotted on the vertical axis,
with time progressing to the right on the horizontal axis.

Such time-series plots trace the behavior of a system through time from
a specific initial state. Thus, such plots represent the behavior of a
system "localized" to a particular initial state, and are referred to as
"local" representations of behavior. In order to get a feeling for the
"global" behavior of a system, behavior independent of any particular
initial state, one can collect an ensemble of such time-series plots,
each rooted at a different initial state, and superimpose them together
in the same plot. For certain systems, such ensembles of local
representations can, in fact, lead to useful insights into the global
dynamics of the system.

However, the "state-space" representation, introduced by Poincare,
provides a much clearer portrait of the global behavior of dynamical
systems. In a state-space representation, the ensemble of all possible
time series is captured in the notion of a vector-field on the state
space: the "field of flow" imposed on the space of states by a
particular dynamical rule. A great deal of insight can be gained into
the behavior of dynamical systems by understanding specific behaviors in
terms of the topological properties of their associated trajectories in
state space.

Although much of the work in the state-space analysis of dynamical
systems has been carried out in the context of continuous state spaces,
many of the concepts and methods carry over to discrete state spaces. In
a discrete state space, the flow field can be seen to be a graph, in
which the states are the nodes and the "flow" is captured by the edges
linking the nodes. Just as one may have fixed points, limit cycles, and
chaotic attractors in continuous flow fields, one may have fixed points,
cycles, and infinite chains in graphs (in the latter case, of course,
the state space must be infinite). Concepts such as the degree of
spreading of a local patch of the flow field in continuous state spaces
have their analogs in the degree of convergence-or "in-degree"-of a node
in the flow graph in discrete state spaces.

The study of Cellular Automata (CA) has proven to be a particularly
rewarding vehicle for gaining insights into the behaviors and
peculiarities of discrete dynamical systems. However, a good deal of the
previous analysis of CA has been carried out via the equivalent of
time-series perspective, in which various properties of the space-time
diagrams of the evolution of CA's from specific initial states are
investigated.

This Atlas presents a comprehensive overview and analysis of CA from the
state-space perspective. Although explicitly treating CA, many of the
observations and results derived here depend only on properties of the
flow graphs themselves, and consequently should be equally valid when
applied to the flow graphs for other discrete dynamical systems.

This Atlas, together with the associated program for generating and
analyzing flow graphs, should prove to be an invaluable tool for
pursuing, in the context of discrete dynamical systems, the kinds of
insights that can only be obtained from a global perspective.

Santa Fe, New Mexico
November 21, 1991

Christopher Langton

Preface - to the First Edition

The study of the dynamical behavior of cellular automata (CA) has become
a significant area of experimental mathematics in recent years. CA
provide a mathematically rigorous framework for a class of discrete
dynamical systems that allow complex, unpredictable behaviour to emerge
from the deterministic local interactions of many simple components
acting in parallel.

Such emergent behavior in complex systems, relying on
distributed rather than centralized control, has become the
accepted paradigm in the attempt to understand biology in terms of
physics (and vice versa?), encompassing such great enigmas as the
phenomena of life and the functioning of the brain. Rather than
confronting these questions head on, an alternative strategy is to pose
the more modest question: how does emergent behaviour arise in CA, one
of the simplest examples of a complex system.

In this book we examine CA behaviour in the context of the global
dynamics of the system, not only the unique trajectory of the system's
future, but also the multiple merging trajectories that could have
constituted the system's past.

In a CA, discrete values assigned to an array of sites change
synchronously in discrete steps over time by the application of simple
local rules. Information structures consisting of propagating
ensembles of values, may emerge within the array, and interact with each
other and with other less active state configurations. Such emergent
behaviour has lead to the notion of computation emerging
spontaneously close to what may be a phase transition in CA
rule space. Emergent behaviour in 2-D CA has given rise to the new field
of artificial life.

In the simpler case of 1-D CA, a trace through time may be made which
completely describes the CA's evolution from a given initial
configuration. This is portrayed as rows of successive global
states of the array, the space-time pattern. Space-time
patterns represent a deterministic sequence of global states evolving
along one particular path within a basin of attraction, familiar
from continuous dynamical systems. In a finite array, the path
inevitably leads to a state cycle. Other sequences of global states
typically exist leading to the same state cycle. The set of all possible
paths make up the basin of attraction. CA basins of attraction are thus
composed of global states linked according to their evolutionary
relationship, and will typically have a topology of branching trees
rooted on attractor cycles.

Other separate basins of attraction typically exist within the set of
all possible array configurations (state space). A CA will, in a
sense, crystallise state space into a set of basins of attraction, known
as the basin of attraction field. The basin of attraction field
is a mathematical object which, if represented as a graph, is an
explicit global portrait of a CA's entire repertoire of behaviour. It
includes all possible apace-time patterns.

The study of basin of attraction fields as a function of CA rule
systems, and how the topology of the fields unfold for increasing array
size, may lead to insights into CA behaviour, and thus to emergent
behaviour in general. This book shows CA basin of attraction fields as
computer graphics diagrams, so that these objects may be as easily
accessible as space-time patterns in experimental mathematics.

Construction of basin of attraction fields poses the problem of finding
the complete set of alternative global states that could have preceded a
given global state, referred to as its pre-images Solving this
problem is recognized as being very difficult, other than by the
exhaustive testing of the entire state space. Exhaustive testing becomes
impractical in terms of computer time as the array size increases beyond
modest values. Consequently, access to these objects has been limited.

This book introduces a reverse algorithm that directly computes
the pre-images of a global state, resulting in an average computational
performance that is many orders of magnitude faster than exhaustive
testing. Two computer programs using the algorithm are described (and
enclosed), to draw either basin of attraction fields or space-time
patterns, for all 1-D, binary, 5-neighbour CA rules, with
periodic boundary conditions, and for the subsets of these rules,
the 3-neighbour rules, and the 5-neighbor totalistic rules.

An atlas is presented (Appendix 2) showing the basin of attraction
fields of all 3-neighbour rules and all 5-neighbour
totalistic rules, produced using the program, for a range of array
lengths. The atlas may be used as an aid to navigation in exploring the
global dynamics of the 232 rules in 5-neighbour rule space.

The book is divided into two parts. The first part (Chapters 1 through 4)
gives the theoretical background and some implications of basin of
attraction fields. The second part consists of appendices including the
atlas and computer-program operating instructions.

Chapter 3 looks in detail at CA architecture and rule systems, and the
corresponding global dynamics. It is shown that ordered
architecture and periodic boundary conditions impose
restrictions on CA evolution in that rotational symmetry (and
bilateral symmetry for symmetrical rules) are
conserved. The rule numbering system and equivalence classes are
reviewed. Symmetry categories, rule clusters, limited
pre-image rules, and the reverse algorithm are
introduced. The Z parameter, which reflects the degree of
preimaging, or the convergence of dynamical flow in state space, is
introduced.

Chapter 4 looks briefly at some implications of the above on current
perceptions of the structure of rule space. The Z parameter is suggested
as the mechanism underlying the lambda parameter . A relationship
between the Z parameter, basin field topology, and rule behaviour
classes, based on the atlas, is proposed.

The idea of the rule table as genotype and the basin of
attraction field as phenotype is examined. Mutating the
rule table is found to result in mutant basin field
topologies. Examples of sets of mutants are presented in Appendix 3.

We hope that the atlas of basin of attraction fields, and the program
for exploring further into rule space, will provide new opportunities
for CA research.

Preface to Second Edition

The "Global Dynamics of Cellular Automata" was published
in 1992 by Addison-Wesley. The publisher was later taken over by Perseus
Books. According to Perseus, in early 1999 all remaining copies of the
book where destroyed. Perseus did this without notifying the authors or the Santa Fe
Institute. Apart from the 1000 plus copies that were sold prior to the
destruction, there are apparently no surviving copies. Normally, the
"remaindered" copies become available when a book goes out of
print.

The copyright was eventually returned to the authors. As there is still
demand for the book, I decided to reprint a Second Edition. The
second edition is a straight copy of one of the original books, with
just a few freehand corrections of typographical and other errors that
have come to light, see the corrections index below. The seven pages of
color plates, which led some people to call the original a "coffee
table book", are omitted to save cost. However, the same reconstructed
images can be seen in color at
www.cogs.susx.ac.uk/users/andywu/.
The original color
cover is also now black and white, with a few changes.

There have been quite a few developments of the ideas first presented in
the book, for example generalizing the methods to Boolean networks,
applications to neural and genetic networks, and methods for classifying
cellular automata automatically. Publications on this work can be found
in the references below. However, many of the book's original ideas,
such as rotation symmetry, rule clusters, limited pre-image rules,
mutations and the rule-space matrix, have not been repeated in other
publications, and others such as the reverse algorithm and the
Z-parameter are explained at length only in the book. Further, the
book is the only place to browse the "Atlas of Basin of Attraction
Fields" itself, so the book remains the only source for most of this
material.

The section "The Atlas Program" remains in the second edition,
describing the DOS software which was included on diskette with the
book. This original software can now be downloaded from
www.santafe.edu/~wuensch/gdca.html. Note that this software has
been superseded by "Discrete Dynamics Lab" (DDLab), a much more
powerful and versatile tool for studying discrete dynamical networks,
running on Unix, Linux and Irix as well as DOS. This is available from
www.ddlab.org.

Many people have commented, why not put the whole book on the web. Good
idea, but as it was produced in the days of cut and paste, that's not
so easy. Anyway, I think its still useful to be able to flip through a
hard copy of "the Atlas''.